Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $φ$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $φ$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log k)^{\frac{1}{2}- o(1)}$ vertices. Further, we prove the existence of an induced path that sees $\frac{k}{2}$ colors.

翻译：给定一个色数为$k$的无三角形图$G$及其一个正常顶点染色$φ$，猜想$G$在$φ$下包含一个长度为$k$的诱导彩虹路径。Scott和Seymour证明了存在长度为$(\log \log \log k)^{\frac{1}{3}- o(1)}$的诱导彩虹路径。我们将此结果改进为存在长度为$(\log k)^{\frac{1}{2}- o(1)}$的诱导彩虹路径。此外，我们证明了存在一条能看见$\frac{k}{2}$种颜色的诱导路径。